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  <section id="module-sympy.physics.quantum.cg">
<span id="clebsch-gordan-coefficients"></span><h1>Clebsch-Gordan Coefficients<a class="headerlink" href="#module-sympy.physics.quantum.cg" title="Permalink to this headline">¶</a></h1>
<p>Clebsch-Gordon Coefficients.</p>
<dl class="py class">
<dt class="sig sig-object py" id="sympy.physics.quantum.cg.CG">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.physics.quantum.cg.</span></span><span class="sig-name descname"><span class="pre">CG</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">j1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j3</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m3</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/physics/quantum/cg.py#L151-L234"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.physics.quantum.cg.CG" title="Permalink to this definition">¶</a></dt>
<dd><p>Class for Clebsch-Gordan coefficient.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>j1, m1, j2, m2</strong> : Number, Symbol</p>
<blockquote>
<div><p>Angular momenta of states 1 and 2.</p>
</div></blockquote>
<p><strong>j3, m3: Number, Symbol</strong></p>
<blockquote>
<div><p>Total angular momentum of the coupled system.</p>
</div></blockquote>
</dd>
</dl>
<p class="rubric">Explanation</p>
<p>Clebsch-Gordan coefficients describe the angular momentum coupling between
two systems. The coefficients give the expansion of a coupled total angular
momentum state and an uncoupled tensor product state. The Clebsch-Gordan
coefficients are defined as <a class="reference internal" href="#r621" id="id1"><span>[R621]</span></a>:</p>
<div class="math notranslate nohighlight">
\[C^{j_3,m_3}_{j_1,m_1,j_2,m_2} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle\]</div>
<p class="rubric">Examples</p>
<p>Define a Clebsch-Gordan coefficient and evaluate its value</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">CG</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span>
<span class="go">CG(3/2, 3/2, 1/2, -1/2, 1, 1)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
<span class="go">sqrt(3)/2</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">CG</span><span class="p">(</span><span class="n">j1</span><span class="o">=</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">m1</span><span class="o">=-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">j2</span><span class="o">=</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">m2</span><span class="o">=+</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">j3</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">m3</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
<span class="go">sqrt(2)/2</span>
</pre></div>
</div>
<p>Compare <a class="reference internal" href="#r622" id="id2"><span>[R622]</span></a>.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Wigner3j</span></code></a></dt><dd><p>Wigner-3j symbols</p>
</dd>
</dl>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r621"><span class="brackets">R621</span><span class="fn-backref">(<a href="#id1">1</a>,<a href="#id3">2</a>)</span></dt>
<dd><p>Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</p>
</dd>
<dt class="label" id="r622"><span class="brackets">R622</span><span class="fn-backref">(<a href="#id2">1</a>,<a href="#id4">2</a>)</span></dt>
<dd><p><a class="reference external" href="https://pdg.lbl.gov/2020/reviews/rpp2020-rev-clebsch-gordan-coefs.pdf">Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions</a>
in P.A. Zyla <em>et al.</em> (Particle Data Group), Prog. Theor. Exp. Phys.
2020, 083C01 (2020).</p>
</dd>
</dl>
</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="sympy.physics.quantum.cg.Wigner3j">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.physics.quantum.cg.</span></span><span class="sig-name descname"><span class="pre">Wigner3j</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">j1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j3</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">m3</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/physics/quantum/cg.py#L28-L148"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner3j" title="Permalink to this definition">¶</a></dt>
<dd><p>Class for the Wigner-3j symbols.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>j1, m1, j2, m2, j3, m3</strong> : Number, Symbol</p>
<blockquote>
<div><p>Terms determining the angular momentum of coupled angular momentum
systems.</p>
</div></blockquote>
</dd>
</dl>
<p class="rubric">Explanation</p>
<p>Wigner 3j-symbols are coefficients determined by the coupling of
two angular momenta. When created, they are expressed as symbolic
quantities that, for numerical parameters, can be evaluated using the
<code class="docutils literal notranslate"><span class="pre">.doit()</span></code> method <a class="reference internal" href="#r623" id="id5"><span>[R623]</span></a>.</p>
<p class="rubric">Examples</p>
<p>Declare a Wigner-3j coefficient and calculate its value</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">Wigner3j</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span> <span class="o">=</span> <span class="n">Wigner3j</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span>
<span class="go">Wigner3j(6, 0, 4, 0, 2, 0)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
<span class="go">sqrt(715)/143</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#sympy.physics.quantum.cg.CG" title="sympy.physics.quantum.cg.CG"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CG</span></code></a></dt><dd><p>Clebsch-Gordan coefficients</p>
</dd>
</dl>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r623"><span class="brackets">R623</span><span class="fn-backref">(<a href="#id5">1</a>,<a href="#id6">2</a>)</span></dt>
<dd><p>Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</p>
</dd>
</dl>
</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="sympy.physics.quantum.cg.Wigner6j">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.physics.quantum.cg.</span></span><span class="sig-name descname"><span class="pre">Wigner6j</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">j1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j12</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j3</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j23</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/physics/quantum/cg.py#L237-L323"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner6j" title="Permalink to this definition">¶</a></dt>
<dd><p>Class for the Wigner-6j symbols</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Wigner3j</span></code></a></dt><dd><p>Wigner-3j symbols</p>
</dd>
</dl>
</div>
</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="sympy.physics.quantum.cg.Wigner9j">
<em class="property"><span class="pre">class</span> </em><span class="sig-prename descclassname"><span class="pre">sympy.physics.quantum.cg.</span></span><span class="sig-name descname"><span class="pre">Wigner9j</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">j1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j12</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j3</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j4</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j34</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j13</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j24</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">j</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/physics/quantum/cg.py#L326-L427"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner9j" title="Permalink to this definition">¶</a></dt>
<dd><p>Class for the Wigner-9j symbols</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Wigner3j</span></code></a></dt><dd><p>Wigner-3j symbols</p>
</dd>
</dl>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="sympy.physics.quantum.cg.cg_simp">
<span class="sig-prename descclassname"><span class="pre">sympy.physics.quantum.cg.</span></span><span class="sig-name descname"><span class="pre">cg_simp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">e</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/physics/quantum/cg.py#L430-L472"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.physics.quantum.cg.cg_simp" title="Permalink to this definition">¶</a></dt>
<dd><p>Simplify and combine CG coefficients.</p>
<p class="rubric">Explanation</p>
<p>This function uses various symmetry and properties of sums and
products of Clebsch-Gordan coefficients to simplify statements
involving these terms <a class="reference internal" href="#r624" id="id7"><span>[R624]</span></a>.</p>
<p class="rubric">Examples</p>
<p>Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to
2*a+1</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">CG</span><span class="p">,</span> <span class="n">cg_simp</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cg_simp</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="n">c</span><span class="p">)</span>
<span class="go">3</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="#sympy.physics.quantum.cg.CG" title="sympy.physics.quantum.cg.CG"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CG</span></code></a></dt><dd><p>Clebsh-Gordan coefficients</p>
</dd>
</dl>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r624"><span class="brackets">R624</span><span class="fn-backref">(<a href="#id7">1</a>,<a href="#id8">2</a>)</span></dt>
<dd><p>Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</p>
</dd>
</dl>
</dd></dl>

</section>


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